As I mentioned in my last post, I recently had a research paper accepted (yay!). The process of writing a paper is both satisfying (I’m accomplishing something) and frustrating (won’t this something ever end?). Perhaps, though, the most annoying thing is that the structure of papers that describe mathematics is so far removed from the production of the same mathematics. For instance, the reader will only see my most polished proofs, not the three less elegant proofs that actually contained the motivation. I like to keep my proofs short, so I usually pull out steps and turn them into lemmas. Part of the justification is that I (or a reader) can then easily cite just that lemma if that is all one needs. This is vastly superior to making people cite the proof of a theorem. However, this is what you end up with:
5 sections laters:
6. Main Theorem
Theorem 2 Wow! What an amazing theorem.
Proof: The only thing cooler than the theorem is its proof. And here is where we use Lemma 1.
There is nothing logically wrong with the order. But how can someone read the lemma and have any reaction other that WTF? Perhaps I add a little remark that the proof of Theorem 2 uses the lemma. This is helpful, but it still feels random.
Now, part of the problem is that I am expecting people to read a paper linearly. The better mathematicians don’t do this; they jump to the good part and only if the idea of the proof doesn’t immediately become transparent do they hop around the paper looking for answers. This, incidentally, is why sticking an important lemma in the preliminary section of a paper can be dangerous: the experts are likely to skip it.
Back on Track
I was reading my favorite group theory text this last weekend and noticed how uneven the exposition is. Early on in the book, the sections jump around from one (seemingly) random topic to another, frequently omitting any discussion of why it is being treated this way or where these topics will show up again. Although the terms below won’t mean much to my readers, I think the point will be obvious. At the end of section 2.7, Gorenstein writes:
The class of [Frobenius groups]…is of fundamental importance in the theory of finite groups, and several basic problems that we shall later investigate stem from this class of groups.
Contrast this with section 2.8 where he writes
The class of [Zassenhaus groups]…is a very important and interesting one, which we shall study in detail in Chapter 13…in a Zassenhaus group, the subgroup fixing a letter is always a Frobenius group in its action on the remaining letters.
Notice that in both remarks he talks about their importance, but that only in the second one does he give an explicit connection to anything. At least with Zassenhaus groups, I know that I can look for more info in Chapter 13 and that they always contain a Frobenius group (which is of fundamental importance in the theory of finite groups, although I don’t have a clue why). I have a bit more to say on this subject, but it probably would behoove me not to get into doubly transitive permutation groups.
No, really. Back on Track.
Historically, we have to write mathematics in such a terse unhelpful way because of two considerations: space and time. Journals don’t want to publish twenty pages of my thoughts just to get 3 pages of mathematical progress. Also, readers (i.e., experts) don’t want to spend an hour wading through explanations of my missteps, wrong turns and false proofs just to get one idea they might already know. On the other hand, people with less experience in the field are likely to wonder at just about every step: why? How in the world did you come up with that unnatural monstrosity? This reader would gladly read some exposition that brings a few things together. They’d also love the reassurance that even experts make mistakes; even better, they would love to see how the experts move from false belief, back to ignorance and finally to enlightenment.
Didn’t I say I had an idea?
The system isn’t going to change any time soon and even if it could, I’m not sure it should. However, perhaps a bit of (would Dan Meyer call it) scaffolding would be useful. Imagine if, in addition to downloading my newest paper off of the arXiv, you could also download a screencast of me going through the paper, telling its story, elaborating on difficult sections and giving hints about how I really understand the subject. I’d be willing to try it myself if I thought anyone would watch, but my research is a bit technical and only a handful of people would follow and even less would care.
So, if nobody cares, why are we still reading?
The thing is, this translates to the relationship between teacher and student. Our students read textbooks, refined over the years to be ruthless, efficient and deadly. The story is missing, the context is missing and the connections are missing. The textbooks are a reference, not a teacher. It is then the teacher’s responsibility to add the missing ingredients, to tell the story, to explain how experts actually think about these things and, most importantly, to teach the students how to read (or understand) a subject non-linearly. Mathematics is structured so poorly K-12 partly because we keep treating the learning of mathematics as a well-ordered system and it isn’t (the axiom of choice, notwithstanding).
Okay, I have so much more to say, but I’ve begun to bore myself. I’ll save the rest for my post on story telling.
Next time: Tricks of the Trade (pt. 3): A Symmetry Trick for Integration